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Theorem m1m1sr 9914
Description: Minus one times minus one is plus one for signed reals. (Contributed by NM, 14-May-1996.) (New usage is discouraged.)
Assertion
Ref Expression
m1m1sr |- ( -1R .R -1R ) = 1R
Proof of Theorem m1m1sr
StepHypRef Expression
1 df-m1r 9884 . . 3 |- -1R = [ <. 1P , ( 1P +P. 1P ) >. ] ~R
21, 1oveq12i 6662 . 2 |- ( -1R .R -1R ) = ( [ <. 1P , ( 1P +P. 1P ) >. ] ~R .R [ <. 1P , ( 1P +P. 1P ) >. ] ~R )
3 df-1r 9883 . . 3 |- 1R = [ <. ( 1P +P. 1P ) , 1P >. ] ~R
4 1pr 9837 . . . . 5 |- 1P e. P.
5 addclpr 9840 . . . . . 6 |- ( ( 1P e. P. /\ 1P e. P. ) -> ( 1P +P. 1P ) e. P. )
64, 4, 5mp2an 708 . . . . 5 |- ( 1P +P. 1P ) e. P.
7 mulsrpr 9897 . . . . 5 |- ( ( ( 1P e. P. /\ ( 1P +P. 1P ) e. P. ) /\ ( 1P e. P. /\ ( 1P +P. 1P ) e. P. ) ) -> ( [ <. 1P , ( 1P +P. 1P ) >. ] ~R .R [ <. 1P , ( 1P +P. 1P ) >. ] ~R ) = [ <. ( ( 1P .P. 1P ) +P. ( ( 1P +P. 1P ) .P. ( 1P +P. 1P ) ) ) , ( ( 1P .P. ( 1P +P. 1P ) ) +P. ( ( 1P +P. 1P ) .P. 1P ) ) >. ] ~R )
84, 6, 4, 6, 7mp4an 709 . . . 4 |- ( [ <. 1P , ( 1P +P. 1P ) >. ] ~R .R [ <. 1P , ( 1P +P. 1P ) >. ] ~R ) = [ <. ( ( 1P .P. 1P ) +P. ( ( 1P +P. 1P ) .P. ( 1P +P. 1P ) ) ) , ( ( 1P .P. ( 1P +P. 1P ) ) +P. ( ( 1P +P. 1P ) .P. 1P ) ) >. ] ~R
9 addasspr 9844 . . . . . 6 |- ( ( 1P +P. 1P ) +P. ( ( 1P .P. ( 1P +P. 1P ) ) +P. ( ( 1P +P. 1P ) .P. 1P ) ) ) = ( 1P +P. ( 1P +P. ( ( 1P .P. ( 1P +P. 1P ) ) +P. ( ( 1P +P. 1P ) .P. 1P ) ) ) )
10 1idpr 9851 . . . . . . . . 9 |- ( 1P e. P. -> ( 1P .P. 1P ) = 1P )
114, 10ax-mp 5 . . . . . . . 8 |- ( 1P .P. 1P ) = 1P
12 distrpr 9850 . . . . . . . . 9 |- ( ( 1P +P. 1P ) .P. ( 1P +P. 1P ) ) = ( ( ( 1P +P. 1P ) .P. 1P ) +P. ( ( 1P +P. 1P ) .P. 1P ) )
13 mulcompr 9845 . . . . . . . . . 10 |- ( 1P .P. ( 1P +P. 1P ) ) = ( ( 1P +P. 1P ) .P. 1P )
1413oveq1i 6660 . . . . . . . . 9 |- ( ( 1P .P. ( 1P +P. 1P ) ) +P. ( ( 1P +P. 1P ) .P. 1P ) ) = ( ( ( 1P +P. 1P ) .P. 1P ) +P. ( ( 1P +P. 1P ) .P. 1P ) )
1512, 14eqtr4i 2647 . . . . . . . 8 |- ( ( 1P +P. 1P ) .P. ( 1P +P. 1P ) ) = ( ( 1P .P. ( 1P +P. 1P ) ) +P. ( ( 1P +P. 1P ) .P. 1P ) )
1611, 15oveq12i 6662 . . . . . . 7 |- ( ( 1P .P. 1P ) +P. ( ( 1P +P. 1P ) .P. ( 1P +P. 1P ) ) ) = ( 1P +P. ( ( 1P .P. ( 1P +P. 1P ) ) +P. ( ( 1P +P. 1P ) .P. 1P ) ) )
1716oveq2i 6661 . . . . . 6 |- ( 1P +P. ( ( 1P .P. 1P ) +P. ( ( 1P +P. 1P ) .P. ( 1P +P. 1P ) ) ) ) = ( 1P +P. ( 1P +P. ( ( 1P .P. ( 1P +P. 1P ) ) +P. ( ( 1P +P. 1P ) .P. 1P ) ) ) )
189, 17eqtr4i 2647 . . . . 5 |- ( ( 1P +P. 1P ) +P. ( ( 1P .P. ( 1P +P. 1P ) ) +P. ( ( 1P +P. 1P ) .P. 1P ) ) ) = ( 1P +P. ( ( 1P .P. 1P ) +P. ( ( 1P +P. 1P ) .P. ( 1P +P. 1P ) ) ) )
19 mulclpr 9842 . . . . . . . 8 |- ( ( 1P e. P. /\ 1P e. P. ) -> ( 1P .P. 1P ) e. P. )
204, 4, 19mp2an 708 . . . . . . 7 |- ( 1P .P. 1P ) e. P.
21 mulclpr 9842 . . . . . . . 8 |- ( ( ( 1P +P. 1P ) e. P. /\ ( 1P +P. 1P ) e. P. ) -> ( ( 1P +P. 1P ) .P. ( 1P +P. 1P ) ) e. P. )
226, 6, 21mp2an 708 . . . . . . 7 |- ( ( 1P +P. 1P ) .P. ( 1P +P. 1P ) ) e. P.
23 addclpr 9840 . . . . . . 7 |- ( ( ( 1P .P. 1P ) e. P. /\ ( ( 1P +P. 1P ) .P. ( 1P +P. 1P ) ) e. P. ) -> ( ( 1P .P. 1P ) +P. ( ( 1P +P. 1P ) .P. ( 1P +P. 1P ) ) ) e. P. )
2420, 22, 23mp2an 708 . . . . . 6 |- ( ( 1P .P. 1P ) +P. ( ( 1P +P. 1P ) .P. ( 1P +P. 1P ) ) ) e. P.
25 mulclpr 9842 . . . . . . . 8 |- ( ( 1P e. P. /\ ( 1P +P. 1P ) e. P. ) -> ( 1P .P. ( 1P +P. 1P ) ) e. P. )
264, 6, 25mp2an 708 . . . . . . 7 |- ( 1P .P. ( 1P +P. 1P ) ) e. P.
27 mulclpr 9842 . . . . . . . 8 |- ( ( ( 1P +P. 1P ) e. P. /\ 1P e. P. ) -> ( ( 1P +P. 1P ) .P. 1P ) e. P. )
286, 4, 27mp2an 708 . . . . . . 7 |- ( ( 1P +P. 1P ) .P. 1P ) e. P.
29 addclpr 9840 . . . . . . 7 |- ( ( ( 1P .P. ( 1P +P. 1P ) ) e. P. /\ ( ( 1P +P. 1P ) .P. 1P ) e. P. ) -> ( ( 1P .P. ( 1P +P. 1P ) ) +P. ( ( 1P +P. 1P ) .P. 1P ) ) e. P. )
3026, 28, 29mp2an 708 . . . . . 6 |- ( ( 1P .P. ( 1P +P. 1P ) ) +P. ( ( 1P +P. 1P ) .P. 1P ) ) e. P.
31 enreceq 9887 . . . . . 6 |- ( ( ( ( 1P +P. 1P ) e. P. /\ 1P e. P. ) /\ ( ( ( 1P .P. 1P ) +P. ( ( 1P +P. 1P ) .P. ( 1P +P. 1P ) ) ) e. P. /\ ( ( 1P .P. ( 1P +P. 1P ) ) +P. ( ( 1P +P. 1P ) .P. 1P ) ) e. P. ) ) -> ( [ <. ( 1P +P. 1P ) , 1P >. ] ~R = [ <. ( ( 1P .P. 1P ) +P. ( ( 1P +P. 1P ) .P. ( 1P +P. 1P ) ) ) , ( ( 1P .P. ( 1P +P. 1P ) ) +P. ( ( 1P +P. 1P ) .P. 1P ) ) >. ] ~R <-> ( ( 1P +P. 1P ) +P. ( ( 1P .P. ( 1P +P. 1P ) ) +P. ( ( 1P +P. 1P ) .P. 1P ) ) ) = ( 1P +P. ( ( 1P .P. 1P ) +P. ( ( 1P +P. 1P ) .P. ( 1P +P. 1P ) ) ) ) ) )
326, 4, 24, 30, 31mp4an 709 . . . . 5 |- ( [ <. ( 1P +P. 1P ) , 1P >. ] ~R = [ <. ( ( 1P .P. 1P ) +P. ( ( 1P +P. 1P ) .P. ( 1P +P. 1P ) ) ) , ( ( 1P .P. ( 1P +P. 1P ) ) +P. ( ( 1P +P. 1P ) .P. 1P ) ) >. ] ~R <-> ( ( 1P +P. 1P ) +P. ( ( 1P .P. ( 1P +P. 1P ) ) +P. ( ( 1P +P. 1P ) .P. 1P ) ) ) = ( 1P +P. ( ( 1P .P. 1P ) +P. ( ( 1P +P. 1P ) .P. ( 1P +P. 1P ) ) ) ) )
3318, 32mpbir 221 . . . 4 |- [ <. ( 1P +P. 1P ) , 1P >. ] ~R = [ <. ( ( 1P .P. 1P ) +P. ( ( 1P +P. 1P ) .P. ( 1P +P. 1P ) ) ) , ( ( 1P .P. ( 1P +P. 1P ) ) +P. ( ( 1P +P. 1P ) .P. 1P ) ) >. ] ~R
348, 33eqtr4i 2647 . . 3 |- ( [ <. 1P , ( 1P +P. 1P ) >. ] ~R .R [ <. 1P , ( 1P +P. 1P ) >. ] ~R ) = [ <. ( 1P +P. 1P ) , 1P >. ] ~R
353, 34eqtr4i 2647 . 2 |- 1R = ( [ <. 1P , ( 1P +P. 1P ) >. ] ~R .R [ <. 1P , ( 1P +P. 1P ) >. ] ~R )
362, 35eqtr4i 2647 1 |- ( -1R .R -1R ) = 1R
Colors of variables: wff setvar class
Syntax hints:   <-> wb 196   = wceq 1483   e. wcel 1990   <.cop 4183  (class class class)co 6650   [cec 7740   P.cnp 9681   1Pc1p 9682   +P. cpp 9683   .P. cmp 9684   ~R cer 9686   1Rc1r 9689   -1Rcm1r 9690   .R cmr 9692
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-omul 7565  df-er 7742  df-ec 7744  df-qs 7748  df-ni 9694  df-pli 9695  df-mi 9696  df-lti 9697  df-plpq 9730  df-mpq 9731  df-ltpq 9732  df-enq 9733  df-nq 9734  df-erq 9735  df-plq 9736  df-mq 9737  df-1nq 9738  df-rq 9739  df-ltnq 9740  df-np 9803  df-1p 9804  df-plp 9805  df-mp 9806  df-ltp 9807  df-enr 9877  df-nr 9878  df-mr 9880  df-1r 9883  df-m1r 9884
This theorem is referenced by:  sqgt0sr  9927
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